I get that the rocket pushes gas out the back and the gas pushes the rocket forward, but if the forces are equal and opposite, shouldn't they cancel out?
The two forces act on different objects, so they never cancel each other out. The rocket pushes hot gas backwards, and the gas pushes the rocket forwards with an equal force. Each object then accelerates according to the force on it, which is why the rocket climbs while the gas shoots down.
Accepted answer
The equal and opposite forces act on two different objects - the exhaust gas and the rocket - so they don't cancel each other out; each object gets its own push. The gas is pushed backward, and by Newton's third law, the rocket is pushed forward with the same-sized force. Since nothing in the rocket+gas system cancels its own force, the rocket accelerates forward.